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Ziolkowski, Marek; Brauer, Hartmut:

Modelling of Seebeck effect in electron beam deep welding of dissimilar metals

URN: urn:nbn:de:gbv:ilm1-2014210107

Published OpenAccess: September 2014

Original published in: Compel : international journal of computation & mathematics in electrical & electronic engineering. - Bradford : Emerald (ISSN-p 0332-1649). - 28 (2009) 1, S. 140-153. DOI: 10.1108/03321640910918940 URL: http://dx.doi.org/10.1108/03321640910918940 [Visited: 2014-09-02]

COMPEL - The international journal for computation and mathematicsin electrical and electronic engineeringModelling of Seebeck effect in electron beam deep welding of dissimilar metalsMarek Ziolkowski Hartmut Brauer

Article information:To cite this document:Marek Ziolkowski Hartmut Brauer, (2009),"Modelling of Seebeck effect in electron beam deep welding ofdissimilar metals", COMPEL - The international journal for computation and mathematics in electrical andelectronic engineering, Vol. 28 Iss 1 pp. 140 - 153Permanent link to this document:http://dx.doi.org/10.1108/03321640910918940

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Modelling of Seebeck effectin electron beam deep welding

of dissimilar metalsMarek Ziolkowski

Department of Electromagnetic Fields/ThEE,Technische Universitat Ilmenau, Ilmenau, Germany andTechnical University of Szczecin, Szczecin, Poland, and

Hartmut BrauerDepartment of Electromagnetic Fields/ThEE,

Technische Universitat Ilmenau, Ilmenau, Germany

Abstract

Purpose – The purpose of this paper is to present a 3D model of deep welding of dissimilar metalsand to show how to model the electron beam deflection due to thermoelectric fields caused bytemperature gradients in some dissimilar metals (Seebeck effect).

Design/methodology/approach – A 3D thermoelectric and heat conduction model is used toestimate the deflection of the electron beam used during welding of dissimilar metals. A weakcoupling between analysed fields is assembled. Additionally, the influence of the deflection on thecalculated fields was not taken into account. The problem is solved using a finite elementmethod.

Findings – It is possible to model Seebeck effect in a relative simple way using the finite elementapproach.

Originality/value – The paper presents a detailed description of modelling procedure of a complexcoupled field problem.

Keywords Electron beam welding, Finite element analysis, Metals, Electrical conductivity

Paper type Research paper

1. IntroductionThe application of highly accelerated electrons as a tool for material processing infusion, drilling and welding processes and also for surface treatment has been knownsince dozen years. The energy conversion in the workpiece indicates that the kineticenergy of the highly accelerated electrons is, at the operational point, not onlyconverted into the heat necessary for welding, but is also released by heat radiationand heat dissipation (Figure 1(a)).

The high-energy density at the impact point causes the evaporation of the metaland thus allowing the following electrons a deeper penetration. This finally leads toa metal vapour cavity which is surrounded by a shell of fluid metal, covering theentire weld depth. This deep-weld effect allows nowadays penetration depths intosteel materials of up to 300mm. When a small diameter electron beam is used tocreate a long joint of two pieces, the beam axis must be in the same plane as thejoint faces and aligned with the joint along its entire length. In the case of twodissimilar metals, even when the beam is properly aligned with the joint, magneticforces can cause beam deflection, resulting in a significant loss of weld quality

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COMPEL28,1

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COMPEL: The International Journalfor Computation and Mathematics inElectrical and Electronic EngineeringVol. 28 No. 1, 2009pp. 140-153q Emerald Group Publishing Limited0332-1649DOI 10.1108/03321640910918940

(Figure 1(b)) (Nazarenko, 1982). One of the reasons of the electron beam deflection isdue to thermoelectric fields caused by temperature gradients in some dissimilarmetals (Seebeck effect (Shercliff, 1979)). Temperature gradients which exist betweenthe top and bottom and in front of and behind the deep and narrow cavityproduced by the electron beam near the joint plane cause thermoelectric currentsflow in the workpiece. These currents induce magnetic field which deflects theelectron beam, even when the beam from the electron gun is properly aligned withthe joint. Interaction between the electron beam and the component of the magneticfield parallel to the welding direction deflects the electron beam in direction ofthe metal with a positive thermoelectric potential (Paulini et al., 1990; Wei andLii, 1990).

2. Seebeck effectThe Seebeck effect can be described by generalized Ohm’s law as:

J ¼ sE2 sS7T; ð1Þ

where J is a conduction current density vector, E is an electric field intensity vector,T denotes temperature, s is an electrical conductivity, and S is called absolutethermoelectric power (Shercliff, 1979). Some data on the absolute thermoelectric powerS of various metals are shown in Figure 2.

In the case of analysis of thermoelectric fields, the electric boundary conditions atinterface between two media of different absolute thermoelectric power must begeneralized and take the form:

Figure 1.Energy conversion duringelectron beam welding (a).

Electron beam deflectionin welding of two

dissimilar metals (b)

Electronbeam

(a) (b)

Thermalradiation

Secondaryelectrons

SolidificationX-ray

Convection

Heat conduction

Fused ZoneKeyhole

V

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J s1

s12

J s2

s2¼ ðS2 2 S1Þ

›T

›s; J n1 ¼ J n2; ð2Þ

in which n and s denote normal and tangential components. Another boundarycondition which is affected by the thermoelectricity is the thermal one which constrainstemperature gradients normal to the interface:

k1›T1

›n2 k2

›T2

›n¼ ðS1 2 S2ÞTJn ¼ ðP1 2P2ÞJ n; ð3Þ

where ki and Pi are the thermal conductivity and Peltier coefficient, respectively.In equations (2) and (3), the contact resistancet is neglected (t ¼ 0). The term (P1-P2)Jndescribes the Peltier effect (thermoelectric cooling), which arises because the ability ofthe current to transport heat changes abruptly at the interface. Since the current mustbe continuous across the interface, the associated heat flow develops a discontinuity ifP1 and P2 are different. This causes heat accumulation or depletion at the interface,depending on the sign of the current.

3. Solution strategyIn the presented paper, we use a 3D thermoelectric and heat conduction model toestimate the deflection of the electron beam used during welding of dissimilar metals.We assume a weak coupling, i.e. the coupling between analysed fields is sequentiallyordered which means that the next calculated field does not have influence on theprevious one. Additionally, we assume that the deflection does not influence calculatedfields, i.e. the deflection is reasonably small. The analysis is split into four followingsteps:

Figure 2.Absolute thermoelectricpower for pure metals

30

sol Fesol Li

sol W

sol Culiq Al

liq Pb

liq Li

sol Pb

sol Pd

sol Nb

liq Na

liq K

liq Hg

Na melts

sol Na

sol Nb

sol Ni

sol K

20

10

0

S [µ

V/K

]

–10

–20

–300 100 200 300 400 500

Temperature [K]Source: Sherceliff (1979)

600 700 800 900

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(1) heat transfer (HT) problem, where the temperature field distribution iscalculated;

(2) conductive media problem, where the current density distribution evoked bySeebeck effect is estimated;

(3) magnetic problem, where the magnetic flux density distribution producedby current density distribution calculated in the previous step is calculated;and

(4) deflection estimation, where the trajectory of the electron beam affected bypreviously calculated magnetic field is found.

Problems 1-3 are solved using a finite element method (FEM). In that case,we have applied the program COMSOL v.3.3 which enables to solve coupledproblems using the same geometry model and can be easy coupled with Matlab.To solve problem 4, we wrote a small procedure in Matlab which simulates thedeflection of the electron beam using a magnetic flux density calculated in FEMprogram.

4. Heat transfer modelDuring the welding process, the electron-beam moves along the weld joint. Thismovement requires a fairly complex model if we want to model the electron-beamas a moving heat source. In the presented analysis, we use a moving coordinatesystem which is fixed at the electron-beam axis and the workpiece is moving inthe opposite direction of x-axis. Performing the coordinate trans-formation, the HTproblem becomes a stationary convection-conduction problem which is much easierto model. The model includes some simplifications, e.g. dimensions of the modelare finite although the transformation demands them infinitely long. It means thatthe analysis does not take into account effects near the start and end of theworkpiece. The model does also not include the stirring process in the fused zone,which is very complex due to phase changes and material flow. The modelgeometry used in the simulations is shown in Figure 3.

The following equation describes HT in the workpiece:

7 · ð2k7TÞ ¼ Qv 2 rCpu ·7T; ð4Þ

where k is thermal conductivity, r is the density, Cp is specific heat capacity, u is thevelocity vector, and Qv denotes a volumetric heat source. The volumetric heat source ismodelled as a conical distribution of power density which has a Gaussian distributionradially and a linear distribution axially (Goldak and Akhlaghi, 2005) and has thefollowing form:

Qv ¼ Q0 1 þ ð1 2 d Þz

h

h ie23ðr=r0Þ

2

; r0 ¼re 2 ðre 2 riÞðhþ zÞ

h; ð5Þ

where Q0 is the maximum value of heat intensity, re, ri, h are dimensions of the keyhole(Figure 3(b)), and d is a parameter which determines the axial distribution of Qv (d ¼ 0linear function, d ¼ 1 constant function).

The maximum heat intensity Q0 can be determined from the absorbedelectron-beam power P using thermal energy conservation law:

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P ¼

V

ZQvdV ¼

Z 0

2h

Z 2p

0

Z r0

0

Qvðr; zÞrdrdwdz: ð6Þ

The final expression of Q0 takes the form:

Q0 ¼36P

pð1 2 e23Þh

1

4ðr2e þ reri þ r2

i Þ2 ð1 2 d Þð3r2e þ 2reri þ r2

i Þ: ð7Þ

Assuming re ¼ ri ¼ r0 and 1 2 e23 ø 1; we can write equation (5) in the followingform:

Qvðr; zÞ ¼6P

pr20h

1 þ ð1 2 d Þz=h

1 þ de23ðr=r0Þ

2

: ð8Þ

Additionally, the following boundary conditions have to be set on the boundaries of theworkpiece (Figure 3(a)):

. left and right walls: thermal insulation:

2n · ð2k7T þ rCpuTÞ ¼ 0;

. back wall: constant temperature T0 which is equal to the ambient temperature(T ¼ T0);

. front wall: convective flux:

q ·n ¼ ðrCpuTÞ ·n; n · ð2k7TÞ ¼ 0;

Figure 3.Model of deep electronbeam welding (a).3D conical Gaussian heatsource (b)

y

Qv

Qv

z

z

xu

Metal 1

Metal 2

(a) (b)

ri

re

h

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. downside and upside walls: they lose heat due to natural convection andsurface-to-ambient radiation. The corresponding heat flux expressions are:

2n · ð2k7T þ rCpuTÞ ¼ hðT0 2 TÞ þ 1sSB T40 2 T 4

� �;

where 1 is hemispherical emissitivity, sSB is Stefan-Boltzman constant, and h is HTcoefficient for natural convection.

5. Conductive media model (DC)Using identity 7 · J ¼ 0 and electric potential definition E ¼ 27V ; we transformequation (1) to the following equation:

7 · ½sðTÞ7V � ¼ 7 · Je; Je ¼ 2sðTÞS7T; ð9Þ

where T is temperature calculated in HT problem. Equation (9) should be solved inregions Metals 1 and 2 (Figure 3). Current density Je is only defined in elements wheretemperature T is less than melting temperature (Tmelt).

On the downside wall, we set potential V ¼ 0 and on all the others, we force anelectric insulation boundary condition ð7 · J ¼ 0Þ: At the interface between metals onlystandard continuity condition n · ðJ 1 2 J 2Þ ¼ 0 is implemented.

6. Magnetic modelfMagnetic flux density B is calculated with the help of magnetic vector potential Adefined as: B ¼ 7 £A; together with Coulomb’s gauge 7 ·A ¼ 0 to assure uniquenessof potential A. Using above and Maxwell equation 7 £H ¼ J; we receive the followingequation:

7 £1

m7 £A

� �¼ J; ð10Þ

where J is the current density distribution calculated in DC problem and m is amagnetic permeability of media. In magnetic model (MA) problem, the geometry of themodel has to be changed by adding additional ambient regions above and below theworkpiece (Figure 3) to enable setting proper boundary conditions for magnetic vectorpotential A. On all side walls, the electric insulation boundary condition ðn £H5 0Þ isset whereas, on other walls, the magnetic insulation boundary condition n £A ¼ 0 ischosen. Magnetic permeability m for both metals is set to mo because even if we haveferromagnetic materials, in most parts of regions where currents evoked by Seebeckeffect are flowing, temperature is greater than the Curie temperature (TCurie) abovewhich ferromagnetic materials loose their magnetic features.

7. Deflection modelThe electron beam trajectory is deflected by Lorentz force according to theequation:

mdv

dt¼ 2ev £ B; ð11Þ

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where m is electron mass (kg), e is electron charge (C), and v is the velocity ofelectrons in the electron beam. Because the Lorentz force is perpendicular to thelocal velocity vector, the magnitude of v remains constant and could be easycalculated from energy conservation law as:

v0 ¼

ffiffiffiffiffiffiffiffiffi2eU

m

r; ð12Þ

where U is accelerating voltage. Displacement r of the electron can be calculatedin an iterative way using following sequence of formulas:

Dv ¼ 2e

mv £ BDt ! vi ¼ vi21 þ Dv; Dr i ¼ viDt! r i ¼ r i21 þ Dri ð13Þ

where Dt is a small enough time step determined by the user. The deflection ofthe electron beam is calculated until the depth of the keyhole is reached.

8. Simulations8.1 Material dataAs Metals 1 and 2, pure iron and copper are chosen, respectively. The thermalconductivity k and the specific heat capacity Cp are defined as temperature dependentfunctions according (Lide, 2006) (Figure 4).

The electrical conductivity is also taken into account as a temperature dependentfunction (Lide, 2006) (Figure 5).

Other material parameters are presented in Table I.

8.2 ResultsIn all simulations, following parameters have been chosen: vector of welding speedu ¼ 22:5ex mm=s;absorbed electron beam power P ¼ 30 kW, radius of the keyholer0 ¼ 0.3 mm, depth of the keyhole h ¼ 70 mm.

Figure 6 shows temperature distribution at various cross-section planes(HT analysis).

To model Seebeck effect, we have applied two approaches:

(1) simplified HT-MA analysis, where J in equation (10) is defined directly as Je inequation (9); and

(2) full HT-DC-MA analysis.

In the first approach, the continuity conditions at the interface between metals are notfulfilled. Figure 7 shows current density distributions for both approaches. Figure 8shows magnetic flux density in the vicinity of fusion zone for MA and DC-MAanalysis.

After calculations of magnetic flux density distributions, the trajectory of theelectron beam has been estimated according to equation (13). We have also implementedthe possibility of tilting the electron beam against the workpiece z-axis whichenables us, in a simple way, to test the influence of the tilt angle on the trajectorybehaviour.

However, both approaches calculate deflection in the proper direction (the electronbeam is deflected towards material with the greater absolute thermoelectric power), we

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Figure 4.Thermal conductivity and

specific heat capacity forpure iron and copper

Thermal conductivity [W/m K]

0

50

100

150

200

250

300

350

400

450

200 400 600 800 1,000 1,200 1,400 1,600 1,800T [K]

Cu

Fe

Specific heat capacity [J/kg K]

300

350

400

450

500

550

600

250 300 350 400 450 500 550 600T [K]

Cu

Fe

Source: Lide (2006)

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can observe that in the case of HT-MA analysis the deflection of the electron beam ismuch stronger compared to HT-DC-MA analysis and the shape of the trajectory is faraway from the observed trajectories in experiments (Figure 1(b)). Therefore, this kindof simplification should be avoided. Figure 9 shows trajectories of the electron beam forHT-MA analysis when the tilt angle is equal to 08 and 58. The trajectory estimationprocedure starts at z ¼ 5 mm above the workpiece surface. In Figure 10, we presentresults of the trajectory estimation for HT-DC-MA analysis. In that case, trajectoriesshapes are more realistic. It was also possible to find the tilt angle which gives almostno deflection of the electron beam.

9. ConclusionsThe presented analysis shows that it is possible to model Seebeck effect in a relativesimple way using FEM. Further investigations should be done to implement thecondition (2) into the analysis and also to study re-coupling of HT and DC analysisthrough the Peltier condition (3).

Figure 5.Electrical conductivity ofpure iron and copper

Electrical conductivity [MS/m]

0

10

20

30

40

50

60

70

250 300 350 400 450 500 550 600 650 700 750 800 850 900T [K]

Cu

Fe

Source: Lide (2006)

Fe Cu

r (kg/m3) 7,870 8,700Tmelt (K) 1,811 1,358S800K (mV/K) 25.0 5.0TCurie (K) 1,043 –

Table I.Parameters of materialsused in simulations

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Figure 6.Temperature distribution

in the vicinity of the fusedzone: (a) plane XY, z ¼ 0;

(b) XYZ view, y ¼ 0;(c) plane YZ, x ¼ 0

u

Fe

Cuy

xz

(a)

uFe

Cu

Fe

Cu

yxz

y xz

(b) (c)

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Figure 7.Current density vectorstogether with temperaturefield distribution in thevicinity of the keyhole forHT-MA (a, c) andHT-DC-MA; (b, d) analysis

u

(a)

(b)

(d)

(c)

Cu

Cu

Fe

FeFe

Cu

Cu

yx

z

yxz

yxz

yx

z

Not

es: (

a)-(

b) p

lane

XY,

z =

–1 m

m, (

c)-(

d) p

lane

YZ,

J a

t x =

0, T

at x

= –

5 m

m

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Figure 8.Magnetic flux densityvectors together with

temperature fielddistribution in the vicinityof the keyhole for HT-MA(a, c) and HT-DC-MA (b, d)

analysis

Fe

CuCu

Cu Cu

Fe

u

z

x yz

x y

zx y

zx y

(a) (b)

(c) (d)

Notes: (a)-(b) plane YZ, B at x = 0, T at x = –5 mm, (c)-(d) plane XZ, B at y = 0, T at y = –5 mm

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Figure 9.Trajectory of electronbeam for HT-MA analysis(a) tilt angle a ¼ 0, (b) tiltangle a ¼ 58

10

0

CuCu

–10

–20

z [m

m]

–30

–40

–50

10

0

–10

–20

z [m

m]

–30

–40

–50

–600 5 10

y [mm](a) (b)

15 0 5 10y [mm]

15

Figure 10.Trajectory of electronbeam for HT-DC-MAanalysis (a) tilt anglea ¼ 0, (b) tilt angle a ¼ 58

10

0

Cu Cu

–10

–20

z [m

m]

–30

–40

–60

–50

10

0

–10

–20

z [m

m]

–30

–40

–50

–600 5 10

y [mm](a) (b)

15 0 5 10y [mm]

15

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References

Goldak, J.A. and Akhlaghi, M. (2005), Computational Welding Mechanics, SpringerScience þ Business Inc., Heidelberg.

Lide, D.R. (2006), CRC Handbook of Chemistry and Physics, 2006-2007, 87th ed.,Taylor&Francis, London.

Nazarenko, O.K. (1982), “Deflection of the electron beam in electron-beam welding”, Avt. Svarka,Vol. 346 No. 1, pp. 33-9.

Paulini, J., Simon, G. and Decker, I. (1990), “Beam deflection in electron beam welding bythermoelectric eddy currents”, J. Phys. D: Appl. Phys., Vol. 23, pp. 486-95.

Shercliff, J.A. (1979), “Thermoelectric magnetohydrodynamics”, J. FluidMech., Vol. 91, pp. 231-51.

Wei, P.S. and Lii, T.W. (1990), “Electron beam deflection when welding dissimilar metals”,Journal of Heat Transfer, Vol. 112, pp. 714-20.

About the authorsMarek Ziolkowski received MSc and PhD in electrical engineering from the Technical Universityof Szczecin in 1978 and 1984. Since 1978 he has been working there at the Chair of TheoreticalElectrotechnics and Computer Science. Since 2001 he is also with the Technische UniversitaetIlmenau. His main research interests include numerical calculations and visualization of EMfields, biomagnetic problems and inverse problems in magnetic fluid dynamics.Marek Ziolkowski is the corresponding author and can be contacted at: marek.ziolkowski@tu-ilmenau.de

Hartmut Brauer received Diploma Engineer and Dr -Ing. in electrical engineering from theIlmenau Institute of Technology in 1975 and 1982. Since 1975 he is with the Department ofElectromagnetic Fields/Theoretical Electrical Engineering at the Technische UniversitaetIlmenau. His main research interests include computation of electromagnetic fields and inversefield problems/optimization in electrical engineering, with applications to bioelectromagnetics,magnetic fluid dynamics and educational aspects.

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